【作者】 徐军；

【导师】 崔云安；

【作者基本信息】 哈尔滨理工大学 ， 基础数学， 2009， 硕士

【摘要】 在这篇论文中,我们主要在Banach空间中引入了几何参数或模,研究了它们的性质及其与一致非方、正规结构、一致正规结构的关系,以及其与不动点之间的联系。本文首先引入U_β-凸模u_β（ε）,并定义U_β-空间,证明了如果存在δ>0,使得u_β（1-δ）>0,则R（X）<2;若u_β（1）>0则X是一致非方的;若存在δ>0,使u_β（1/2-δ）>0,则X具有正规结构,从而X具有不动点性质。接着讨论了由J.Gao在最近引入的一个二次常数E（X）。另外还得到了刻画空间超自反性的一个新的条件:X超自反（?）（E|^）（X）<8。从而得到了Banach空间有不动点性质的三个充分条件。然后讨论了Clarkson凸性模的推广形式,即广义凸性模δ^α_X（ε）的一些几何性质。另外还得到了广义凸性模的一些函数性质如单调性、连续性等。进而又得到了空间有正规结构的充分条件,即若存在ε,0≤ε≤1,使得δX（α）（1+ε）>（1-α）ε,则X具有正规结构;假如存在ε∈[0,1]和α∈[0,1]满足δX（α）（1+ε）>f（ε）,那么X具有正规结构。这些结果都推广了J.Gao的结论。最后我们还利用光滑模和非方常数得到了Banach空间上的集值非扩张映射存在不动点的一个充分条件。更多还原

【Abstract】 Geometric parameters or moduli of a Banach space are mainly introduced in this thesis. Their properties and the relationship among uniform nonsquareness, normal structure and uniform normal structure are studied, and the contact with fixed points is also established.TheU_β- modulus of convexity u_β（ε）andU_β- space are defined. We prove that if there existsδ> 0, such that u_β（1 -δ）> 0, then R （ X ） < 2; We also obtain the results that if u_β（1） > 0, then X is uniformly nonsquare, and that if there isδ> 0, such that u_β（1 / 2 -δ）> 0, then X has normal structure, and therefore X has the fixed point property.Next, the coefficient E （ X ）, introduced by J.Gao recently, is also studied. Furthermore, a sufficient and necessary condition, which imply a Banach space X is super reflexive is equivalent to （E|-）（X） < 8, is given.Generalizing all the previous results, and three sufficient conditions for Banach space X to have fixed point property are obtained.Followed that, the generalization of Clarkson’s moduli of convexity is investigated. Those are some geometric properties ofδ^α_X（ε）. The functional properties of this modulus, such as monotonity and continuity, are also obtained. Moreover, some sufficient conditions for X to have normal structure are obtained, that is, if there exists anε, 0≤ε≤1, such thatδ^α_X （1 +ε） > （1 -α）ε, then X has a normal structure; If there exists anε∈[0,1] and anα∈[0,1], such thatδ^α_X（1 +ε） > f（ε）, then X has a normal structure. These results generalize the conclution of J.Gao.Finally, a sufficient condition is achieved that nonexpansive set-valued mappings have fixed points on Banach space by means of moduli of smoothness nonsquare constants.更多还原